4 Elementary Functions Applications 4.42 Solution of Triangles 4.44 Other Applications

§4.43 Cubic Equations

ⓘ

Keywords:

applications, cubic equations, solutions as trigonometric and hyperbolic functions, trigonometric functions

Notes:

See Hobson (1928, p. 335).

Referenced by:

§1.11(iii), §3.8(iv), Erratum (V1.0.10) for Section 4.43

Permalink:

http://dlmf.nist.gov/4.43

Errata (effective with 1.0.10):

The first paragraph of this section has been rewritten to correct reported errors. The constants

C=(-\frac{27q^{2}}{4p^{3}})^{1/2}

and

D=-(\frac{27q^{2}}{4p^{3}})^{1/2}

, formerly given in Eq. (4.43.1) for use in Cases (a), (b), and (c) of Eq. (4.43.2), have been eliminated, and these three cases have been corrected. Formerly they read

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=C$ , when $p<0$ and $C\leq 1$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=C$ , when $p<0$ and $C>1$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=D$ , when $p>0$ .

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for Ch.4

Let $p(\neq 0)$ and $q$ be real constants and

4.43.1	$\displaystyle A$	$\displaystyle=\left(-\tfrac{4}{3}p\right)^{1/2},$
4.43.1	$\displaystyle B$	$\displaystyle=\left(\tfrac{4}{3}p\right)^{1/2}.$
ⓘ Symbols: $p$ : real constant and $q$ : real constant Referenced by: (4.43.2), §4.43 Permalink: http://dlmf.nist.gov/4.43.E1 Encodings: TeX, TeX, pMML, pMML, png, png Errata (effective with 1.0.10): The constants $C=(-\frac{27q^{2}}{4p^{3}})^{1/2}$ and $D=-(\frac{27q^{2}}{4p^{3}})^{1/2}$ , formerly given in this equation for use in Cases (a), (b), and (c) of Eq. (4.43.2), have been eliminated, and these three cases have been corrected. Reported 2014-10-31 by Masataka Urago See also: Annotations for §4.43 and Ch.4

The roots of

4.43.2		$z^{3}+pz+q=0$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant, $z$ : complex variable, $p$ : real constant and $q$ : real constant Referenced by: (4.43.1), §4.43 Permalink: http://dlmf.nist.gov/4.43.E2 Encodings: TeX, pMML, png Errata (effective with 1.0.10): Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read (a) $A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=C$ , when $p<0$ and $C\leq 1$ . (b) $A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=C$ , when $p<0$ and $C>1$ . (c) $B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=D$ , when $p>0$ . Reported 2014-10-31 by Masataka Urago See also: Annotations for §4.43 and Ch.4

are:

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=4q/A^{3}$ , when $4p^{3}+27q^{2}\leq 0$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=-4q/A^{3}$ , when $p<0$ , $q<0$ , and $4p^{3}+27q^{2}>0$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=-4q/B^{3}$ , when $p>0$ .

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

4.42 Solution of Triangles 4.44 Other Applications

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§4.43 Cubic Equations

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